Ocean Wave Spectra and Shtp Appltcattons 
era (2. 4) 
8 -ik Y lwT 
| i e : S(k 5) dk e dw 
— 00 
g 
where S(k,; )dk, dw=S(kK) dk. The dependence of S on k, and 
k, is replaced here by dependence on k, and , making use of the 
dispersion relation between k, > 7 andw. Equation (2.4) relates 
the directional wave spectrum S (k’) to the correlation function 
R(7,7%) . In the above equation 
Fourier transform of R(7, £) with respect to 7 and is called the 
cross spectrum, which is denoted by Co() - iQ(w). Thus, 
RNG 7) = / [Co(w5¥) - iQ(wi¥)] aT ay (2. 5) 
where ih 
o/s = = 
| S(k 5) oo “dk, = Co(w3t) - iQ(w:t) 
=" /g (2. 6) 
Thus, if one represents the directional wave spectrum S(k,,w) ina 
Fourier series with respect to k,D, where D satisfies _w2h Sm 4; 
one can obtain the Fourier coefficients directly from equatfon (2. 6) 
by their definitions. That is 
3D 
S(k), #) = Pann ) : (A, cos nk, D+B_ sin nk D) (2-4) 
ra ol 
where 
A. ==— Co(w3? = (0,0) ) 
(2. 8) 
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